Prediction, Belief, and Markets

Transcription

1 Prediction, Belief, and Markets Jake Abernethy, UPenn è UMich Jenn Wortman Vaughan, MSR NYC July 14, 2013

2 Belief, Prediction, and Gambling? A Short History Lesson The earliest references to probability calculations arose directly through the study of games of chance, like dice. We have evidence of human gambling as early as ancient Egypt, more than 3000 years ago. We have no evidence of probabilistic analysis until the time Pascal in the late 1600s.

3 The Emergence of Probability

4 Probability and Finance: It s Only a Game!

5 Letters b/t Pascal and Fermat, 1654 The beginning of mathematical probability is often dated to letters between Pierre de Fermat and Blaise Pascal. The setting discussed by Pierre and Fermat: two players are playing a game, they need to get n points to win, and the winner receives prize. How to divide the prize if the game is cut short?

6 Bruno de Finetti: PROBABILITY DOES NOT EXIST De Finetti, well-known in probability theory, had the view that we may only think about probabilities in terms of rates of betting. That is, the laws of probability can be viewed as resulting from simple no-arbitrage conditions on these rates

7 Betting can be used to elicit beliefs Economists, in particular, are very keen on betting with each other. Alex Tabarrok, in marginalrevolution.com: A Bet is a Tax on Bullshit There s been some recent debate in the Econ blogosphere about the extent to which bets really do reveal beliefs. Noah Smith, of noahpinion.com: The mistake is looking at the risk and return of single assets instead of portfolios. Basically, the risk of an asset is based mainly on how that asset related to other assets in your portfolio.

8 How do I find someone to bet with? Answer: A Prediction Market Prediction markets have existed for over 200 years. Typically, odds (prices) are set by supply and demand. People began to notice: the market prices are generally very accurate, and provide better predictors than expert assessments, etc. What s going on? Robin Hanson: Rational expectations theory predicts that, in equilibrium, asset prices will reflect all of the information held by market participants. This theorized information aggregation property of prices has lead economists to become increasingly interested in using securities markets to predict future events.

9 Outline Before the break: 1. Predictions Markets in Practice 2. Eliciting beliefs with proper scoring rules 3. Bregman divergences + proper scoring rules 4. Hanson s Market Scoring Rule After the break: 1. Securities markets 2. Duality & connections to online learning 3. Handling very large outcome spaces 4. Overview of additional topics

10 The Most (in)famous Prediction Market 1999: Intrade founded by John Delaney 2003: Acquired by TradeSports in 2003, later splits off after TradeSports closes down in : Intrade gains notoriety during Bush/Kerry election for providing continuous forecasts throughout campaign May 2011: Founder John Delaney dies at age 42 while climbing Mt. Everest, less than 50 meters from summit Nov. 2012: US regulator CFTC files suit against Intrade, leading Intrade to disallow US customers from betting Mar. 2013: Due to financial irregularities, Intrade halts trading, freezes all accounts. Still remains in legal limbo.

11 Example: Intrade

12 Iowa Electronic Markets (IEM): Legal and with Real Money Founded in 1988 at the University of Iowa for the purpose of research in market prediction accuracy Received a no action letter from the CFTC, permitting them to facilitate unregulated betting. (Such letters are apparently no longer being given out ) On the downside, the IEM must obey a certain set of conditions. Most notably, individual traders may deposit no more than $500.

13 Predictious: A New Bitcoin-based Prediction Market

14 Example: Inkling Markets Internal prediction markets used within companies

15 Markets in Practice Questions: 1. What are different market mechanisms? 2. How quickly do markets incorporate information? 3. How accurate are market prices, vis-à-vis prediction?

16 Arrow-Debreu Securities Arrow-Debreu Security : Contract pays $10 if X happens, $0 otherwise. If I think that Pr(X) = p then I should: Buy this security at any price less than $10p Sell this security at any price greater than $10p Current price measures the population s collective beliefs

17 [1] Market Mechanisms: Continuous Double Auction (CDA) Used by Intrade.com and Betfair.com Market receives a sequence of orders Two types of orders: Limit order: trader posts shares to order book Market order: trader buys shares in order book

18 Obama2012 Intrade: Bid+Ask+Trades

19 Aside: Problems with the CDA Chicken and egg problem: who is willing to join a market if there are no other participants? Not a lot of liquidty : it s very easy to swing prices Large bid/ask spreads Alternative mechanism: the automated market maker, which we will be discussing throughout the 2 nd half of the tutorial

20 [2] How Quickly do Markets Respond? Source: Snowberg, Wolfers, Zitzewitz 2012

21 [3] Are Market Prices Accurate? The market price for Arrow-Debreu security is essentially a consensus estimate of the probability of an event Are these estimates accurate? We can check this on historical data, but Prices are changing, which price do we use? What is the right metric to measure accuracy? What are we comparing against?

22 Market Prediction vs. True Vote Share Berg et al., 2008: Results From a Dozen Years of Election Futures Markets Research

23 Average Polls vs. Market Prices Poll Error: (average from last week) Market Error: (election eve) Market Error: (average from last week) 1.91% 1.49% 1.58% Berg et al., 2008: Results From a Dozen Years of Election Futures Markets Research

24 Aside: Supreme Court + Health Care Intrade market: The US Supreme Court to rule individual mandate unconstitutional before midnight ET 31 Dec 2012

25 More on Obamacare Prediction Market David Leonhardt in the NYTimes: After several years in which the market was often celebrated as a crystal ball, the Supreme Court ruling was a useful corrective. The prediction-market revolution, like so many others, initially promised more than it could deliver. Response by Robin Hanson on overcomingbias.com: But the Intrade market on the Obamacare court case was an active valid market, on an appropriate subject. When it assigned a 75% chance to an event it was saying real loud that it would be wrong 1/4 of the time. And studies have consistently found such markets are well-calibrated in this way. What more do you want?

26 The Basics: Proper Scoring Rules

27 1950: Brier on Weather Forecasting

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30 How Should We Pay a Forecaster? What is the right payment scheme to reward/punish a forecaster who makes a sequence of probability predictions for events that we observe? The sequence of outcomes: The sequence of forecasts: The forecaster s payment: y 1, y 2, y 3,... {0,1} p 1, p 2, p 3,... [0,1] 1 T T t=1 S(y t, p t )

31 Brier Score ó Quadratic Scoring Rule For a binary outcome y {0,1}, p [0,1] S(y, p) = (y p) 2 For one of n outcomes, y {1,...,n}, p n n i=1 S(y, p) = (1 y=i p i ) 2

32 What s Special About This Function? S(y, p) = (y p) 2 Assume y is random and Pr(y = 1) = q. Then... argmax Ε (y p) 2 p [0,1] ( [ ] ) = = argmax( q(1 p) 2 (1 q) p 2 ) p [0,1] argmax( (p q) 2 q + q 2 ) = q p [0,1]

33 Proper Scoring Rules What we have just introduced is the notion of a proper scoring rule, any function S satisfying Ε y ~q [ S(y,q) ] Ε [ S(y,p) ] p,q Δ y ~q n The scoring rule is said to be strictly proper if the above inequality is strict unless p = q

34 Another Strictly Proper Scoring Rule S(y,p) = log p(y) This is known as the logarithmic scoring rule. For binary random variables, it can be written as: " $ S(y, p) = # %$ log p y =1 log(1 p) y = 0 EXERCISE: check that this is proper!

35 Scoring Rules == Loss Functions? Effectively, a scoring rule is just a type of loss function Scoring rules measure the performance (not loss) of a predicted distribution given a final outcome Research on scoring rules is focused more heavily on the incentives of the associated payment mechanism

36 Designing Scoring Rules Using Bregman Divergences

37 Savage 1973

38 Digression: Bregman Divergences A Bregman divergence measures distance with respect to a convex function f D f ( p,q) = f ( p) f (q) f (q) ( p q)

39 Digression: Bregman Divergences D f ( p,q) = f ( p) f (q) f (q) ( p q) Properties: D f ( p, p) = 0 D f ( p,q) 0 p p,q D f ( p,q) D f (q, p) (in general)

40 Bregman Divergences III D f ( p,q) = f ( p) f (q) f (q) ( p q) Example 1, quadratic: f ( p) = p 2 D f (p,q) = p q 2 Example 2, entropic: f (p) = p log p D (p, q) = p log p i i i f i i i q i

41 Bregman Diverg.ó Scoring Rule?? Let f be any convex function Let e i be the ith indicator vector, e i = 0,,0,1,0,,0 Let p, q be any two distributions Fact: There exists a function g such that E i~q [ D (e,p)] = D (q,p) + g(q) f i f and so argmaxe [ i~q D f (e i,p)] = q p Δ n This is the scoring rule property!!

42 Bregman Diverg.ó Scoring Rule!! We now have a recipe for constructing scoring rules: Take any convex function f and set S(i,p) = D f (e i,p) Quadratic Scoring Rule: f (p) = p 2 2 Log Scoring Rule: f (p) = i p i log p i

43 Brief Literature Review

44 Market Scoring Rules for Belief Aggregation

45 Learning a Consensus? Scoring rules are useful for incentivizing one individual to state his beliefs about a probability, but what if we d like to learn from a crowd Proposal: We could just pay every individual according to a scoring rule. Problems: This could be very expensive! How should we combine estimates? How can we weed out noise traders? How can we weed out copycats?

46 Market Scoring Rules Robin Hanson proposed the following idea to create a prediction market based on an automated market maker: Suppose we have a random variable X which will take one of n values {1, 2,, n} The MM chooses a scoring rule S and announces it The MM then posts an initial distribution (prior) p 0 Traders arrive, one-by-one, giving updates p t-1 p t Eventually, outcome X is revealed, and trader t earns (or loses) S(X, p t ) S(X, p t 1 )

47 Market Scoring Rule A p1 p0 p2 B pt C

48 Incentives and Costs Assume trader t has belief distribution p on X, which can (and should!) depend on previous market observations Suppose he wants to maximize his payment argmax pt Ε X~p [S(X, p t ) S(X, p t 1 )] The MM must make all payments, which total T [ S(X,p t ) S(X,p t 1 )] = S(X,p T ) S(X,p 0 ) t =1 = argmax always pt Ε X~p [S(X, non-negative! p t )] = p This is bounded! This is like MM s subsidy to market.

49 LMSR: Log Market Scoring Rule Initial hypothesis p0 is the uniform distribution Trader t posts an update pt-1 pt After X is revealed, trader t earns log(pt(x)/pt-1(x)) Hanson: the LMSR is an important special case, the only MSR for which betting on conditional probabilities does not affect marginal probabilities The market maker s worst case loss is bounded by log n, where n is the number of possible values of X

50 Prediction, Belief, and Markets: Part 2 Jake Abernethy, UPenn UMich Jenn Wortman Vaughan, Microsoft Research NYC

51 Outline of Part 2 1. Automated market makers for securities markets 2. Designing markets for large outcome spaces 3. Duality & connections to online learning 4. Recent extensions and additional topics

52 Back to Securities Markets

53 Arrow Debreu Securities

54 Arrow Debreu Securities Potential payoff is $10. If I think that the probability of this event is p, I should Buy this security at any price less than $10p Sell this security at any price greater than $10p Current price measures the population s collective beliefs

55 Arrow Debreu Securities How do we arrive at the current price?

56 Arrow Debreu Securities How do we arrive at the current price? Traditional stock market style pricing (continuous double auction) low liquidity, huge spreads

57 Arrow Debreu Securities How do we arrive at the current price? Traditional stock market style pricing (continuous double auction) low liquidity, huge spreads Automated market maker willing to risk a (bounded) loss in order to encourage trades

58 Market Makers for Complete Markets In a complete market, a security is offered for each of a set of mutually exclusive and exhaustive events

59 Market Makers for Complete Markets In a complete market, a security is offered for each of a set of mutually exclusive and exhaustive events Worth $1 iff Worth $1 iff

60 Market Makers for Complete Markets In a complete market, a security is offered for each of a set of mutually exclusive and exhaustive events Worth $1 iff Worth $1 iff An automated market maker is always willing to buy and sell these securities at some price

61 Cost Functions

62 Cost Functions Worth $1 iff Worth $1 iff Already purchased: q 1 shares q 2 shares

63 Cost Functions Worth $1 iff Worth $1 iff Already purchased: q 1 shares q 2 shares Want to purchase: r 1 shares r 2 shares

64 Cost Functions Cost of purchase: C(q + r) C(q) Worth $1 iff Worth $1 iff Already purchased: q 1 shares q 2 shares Want to purchase: r 1 shares r 2 shares

65 Cost Functions Cost of purchase: C(q + r) C(q) Worth $1 iff Worth $1 iff Already purchased: q 1 shares q 2 shares Want to purchase: r 1 shares r 2 shares Instantaneous prices: p 1 = C / q 1 p 2 = C / q 2

66 Cost Functions Cost of purchase: C(q + r) C(q) Worth $1 iff Worth $1 iff Already purchased: q 1 shares q 2 shares Want to purchase: r 1 shares r 2 shares Instantaneous prices: p 1 = C / q 1 p 2 = C / q 2 predictions

67 Back to the LMSR Remember the logarithmic market scoring rule Initial hypothesis p0 is the uniform distribution Trader t posts an update pt-1 pt After outcome i is revealed, trader t receives log(pt,i) log(pt-1,i) = log(pt,i / pt-1,i)

68 The logarithmic market scoring rule can be implemented as a cost function based market with cost function and instantaneous prices Back to the LMSR C(q1,...,qN) = log p i = exp(q i ) N i = 1 Σ j exp(q j ) exp(qi)

69 The logarithmic market scoring rule can be implemented as a cost function based market with cost function and instantaneous prices Back to the LMSR C(q1,...,qN) = log p i = exp(q i ) i = 1 Σ j exp(q j ) exp(qi) Notice that p i is increasing in q i and the prices sum to 1 N

70 Equivalence For all p, p', q, q', such that C(q) = p and C(q') = p', for all outcomes i, a trader who changed the market state from p to p' in the MSR would receive the same total payoff as a trader who changed the market state from q to q' in the cost function based market. [Hanson 03; Chen & Pennock 07]

71 A Proof in One Slide

72 cost function payoff A Proof in One Slide = ( q' i q ) i ( C(q') C(q) ) security value when the outcome i occurs cost of the purchase

73 cost function payoff A Proof in One Slide = ( q' i q ) i ( C(q') C(q) )

74 cost function payoff A Proof in One Slide ( ) C(q') C(q) = q' i q i ( ) log e q' j & = q' i q i ( ) $ & % log e q j j j ' ) ( by definition

75 cost function payoff A Proof in One Slide ( ) C(q') C(q) = q' i q i ( ) log e q' j & = q' i q i ( ) $ & % log e q j j j ' ) (

76 cost function payoff A Proof in One Slide = ( q' i q ) i ( C(q') C(q) ) ( ) log e q' j & = loge q' i loge q i $ & % log e q j j j ' ) (

77 cost function payoff A Proof in One Slide = ( q' i q ) i ( C(q') C(q) ) ( ) log e q' j & = loge q' i loge q i = log eq' i e log e q' j j j $ & % q i e q j log e q j j j ' ) (

78 cost function payoff A Proof in One Slide = ( q' i q ) i ( C(q') C(q) ) ( ) log e q' j & = loge q' i loge q i = log eq' i e log e q' j j j $ & % q i e q j log e q j j j prices! ' ) (

79 cost function payoff A Proof in One Slide = ( q' i q ) i ( C(q') C(q) ) ( ) log e q' j & = loge q' i loge q i = log eq' i e log e q' j j j $ & % q i e q j log e q j j j ' ) ( = log p' i log p i

80 cost function payoff A Proof in One Slide = ( q' i q ) i ( C(q') C(q) ) ( ) log e q' j & = loge q' i loge q i = log eq' i e log e q' j j j $ & % q i e q j log e q j j j ' ) ( = log p' i log p i = scoring rule payoff

81 More Generally Any market scoring rule can be implemented as a cost function based market [Chen & Pennock 07; Chen & Vaughan 10; Abernethy & Frongillo 11; ]

82 More Generally Any market scoring rule can be implemented as a cost function based market [Chen & Pennock 07; Chen & Vaughan 10; Abernethy & Frongillo 11; ] Advantages: Retains the good incentive properties of the MSR Arguably more natural for traders Exposure to risk is more transparent

83 Beyond Complete Markets

84 Complex Outcome Spaces n! 2 n Infinite

85 Complex Outcome Spaces n! 2 n Infinite MSR-NYC s WiseQ [Dudik et al., 2013]

86 Complex Outcome Spaces n! 2 n Infinite Cannot simply run a standard market like LMSR Calculating prices is intractable [Chen et al., 2008] Reasoning about probabilities is too hard for traders

87 Complex Outcome Spaces n! 2 n Infinite Cannot simply run a standard market like LMSR Calculating prices is intractable [Chen et al., 2008] Reasoning about probabilities is too hard for traders Can run separate, independent markets (e.g., horses to win, place, or show) but this ignores logical dependences

88 Complex Outcome Spaces Given a small set of securities over a very large (or infinite) state space, how can we design a consistent market that can be operated efficiently? [Abernethy, Chen, and Vaughan, EC 2011; long version in ACM TEAC 2013]

89 Menu of Securities We would like to offer a menu of securities{1,, K} specified by a payoff function ρ

90 Menu of Securities We would like to offer a menu of securities{1,, K} specified by a payoff function ρ payoff securities outcomes

91 Example: Pair Betting $1 if and only if horse i finishes ahead of horse j

92 Example: Pair Betting $1 if and only if horse i finishes ahead of horse j A<B B<A A<C C<A B<C C<B ABC ACB BAC BCA CAB CBA

93 What are reasonable prices?

94 What are reasonable prices? For complete markets

95 What are reasonable prices? i For complete markets p i =1

96 What are reasonable prices? For complete markets i p i =1 For pair betting

97 What are reasonable prices? For complete markets i p i =1 For pair betting p i< j + p j<i =1

98 What are reasonable prices? For complete markets i p i =1 For pair betting p i< j + p j<i =1 1 p i< j + p j<k + p k<i 2

99 What are reasonable prices? For complete markets i p i =1 For pair betting p i< j + p j<i =1 1 p i< j + p j<k + p k<i 2 what else?

100 What are reasonable prices? For complete markets i p i =1 For pair betting p i< j + p j<i =1 1 p i< j + p j<k + p k<i 2 what else? In general???

101 An Axiomatic Approach Path independence: The cost of acquiring a bundle r of securities must be the same no matter how the trader splits up the purchase.

102 An Axiomatic Approach Path independence: The cost of acquiring a bundle r of securities must be the same no matter how the trader splits up the purchase. Formally, Cost(r + r r 1, r 2,, r t ) = Cost(r r 1, r 2,, r t ) + Cost(r r 1, r 2,, r t, r)

103 An Axiomatic Approach Path independence: The cost of acquiring a bundle r of securities must be the same no matter how the trader splits up the purchase. Formally, Cost(r + r r 1, r 2,, r t ) = Cost(r r 1, r 2,, r t ) + Cost(r r 1, r 2,, r t, r) This alone implies the existence of a cost potential function! Cost(r r 1, r 2,, r t ) = C(r 1 + r r t + r) C(r 1 + r r t )

104 An Axiomatic Approach Existence of instantaneous prices: C must be continuous and differentiable

105 An Axiomatic Approach Existence of instantaneous prices: C must be continuous and differentiable Information incorporation: The purchase of a bundle r should never cause the price of r to decrease

106 An Axiomatic Approach Existence of instantaneous prices: C must be continuous and differentiable Information incorporation: The purchase of a bundle r should never cause the price of r to decrease No arbitrage: It is never possible to purchase a bundle r with a guaranteed positive profit regardless of outcome

107 An Axiomatic Approach Existence of instantaneous prices: C must be continuous and differentiable Information incorporation: The purchase of a bundle r should never cause the price of r to decrease No arbitrage: It is never possible to purchase a bundle r with a guaranteed positive profit regardless of outcome Expressiveness: A trader must always be able to set the market prices to reflect his beliefs

108 An Axiomatic Approach Theorem: Under these five conditions, costs must be determined by a convex cost function C such that { C(q) : q R K } Hull(ρ)

109 [ An Axiomatic Approach Theorem: Under these five conditions, costs must be determined by a convex cost function C such that reachable price vectors { C(q) : q R K } Hull(ρ)

110 [ An Axiomatic Approach Theorem: Under these five conditions, costs must be determined by a convex cost function C such that reachable price vectors outcomes { C(q) : q R K } Hull(ρ) securities

111 [ An Axiomatic Approach Theorem: Under these five conditions, costs must be determined by a convex cost function C such that reachable price vectors outcomes { C(q) : q R K } Hull(ρ) securities

112 Cost Functions Via Duality & The Connection to Online Learning

113 How do we find these cost functions? Fact: A closed, differentiable function C is convex if and only if it can be written in the form C(q) = sup x q R(x) x dom(r) for a strictly convex function R called the conjugate.

114 How do we find these cost functions? Fact: A closed, differentiable function C is convex if and only if it can be written in the form C(q) = sup x q R(x) x dom(r) for a strictly convex function R called the conjugate. Furthermore, C(q) = arg max x q R(x) x dom(r)

115 How do we find these cost functions? Fact: A closed, differentiable function C is convex if and only if it can be written in the form C(q) = sup x q R(x) x dom(r) for a strictly convex function R called the conjugate. Furthermore, C(q) = arg max x q R(x) x dom(r) To generate a convex cost function C, we just have to choose an appropriate conjugate function and domain!

116 But how do we choose R?

117 But how do we choose R? We can borrow ideas from online linear optimization (or the simpler expert advice setting) and in particular, Follow the Regularized Leader algorithms Market s conjugate function regularizer

118 Learning from Expert Advice Suppose we would like to choose actions based on the advice of n experts (people, algorithms, features )

119 Learning from Expert Advice Suppose we would like to choose actions based on the advice of n experts (people, algorithms, features )

120 Learning from Expert Advice Suppose we would like to choose actions based on the advice of n experts (people, algorithms, features ) At each round t, Algorithm selects weights w t,1 w t,2 w t,3

121 Learning from Expert Advice Suppose we would like to choose actions based on the advice of n experts (people, algorithms, features ) At each round t, Algorithm selects weights Experts suffer a loss w t,1 w t,2 w t,3 l t,1 l t,2 l t,3

122 Learning from Expert Advice Suppose we would like to choose actions based on the advice of n experts (people, algorithms, features ) At each round t, Algorithm selects weights Experts suffer a loss Algorithm suffers a loss w t,1 w t,2 w t,3 l t,1 l t,2 l t,3 w t l t

123 What is the goal? Ideally, we d like to bound the cumulative loss T t =1 w t l t

124 What is the goal? Ideally, we d like to bound the cumulative loss T t =1 w t l t Instead, we look at the algorithm s regret T t =1 w t l t min w K w L T

125 What is the goal? Ideally, we d like to bound the cumulative loss T t =1 w t l t Instead, we look at the algorithm s regret T t =1 w t l t min w K w L T algorithm s loss

126 What is the goal? Ideally, we d like to bound the cumulative loss T t =1 w t l t Instead, we look at the algorithm s regret T t =1 w t l t min w K w L T cumulative loss vector algorithm s loss loss of the best fixed weight vector in hindsight

127 What is the goal? Ideally, we d like to bound the cumulative loss T t =1 w t l t Instead, we look at the algorithm s regret T t =1 w t l t min w K w L T

128 What is the goal? Ideally, we d like to bound the cumulative loss T t =1 w t l t Instead, we look at the algorithm s regret T t =1 w t l t min w K w L T Can achieve optimal (O(T ½ )) regret with Follow the Regularized Leader w t+1 = argmin w K w L t + R(w) cumulative loss regularizer

129 Online Linear Opt. Market Making

130 Online Linear Opt. Learner maintains weights w t K over n items/experts Market Making Market maker maintains prices p t Π over n contracts

131 Online Linear Opt. Learner maintains weights w t K over n items/experts Items have loss vector l t, cumulatively L t+1 = L t + l t+1 Market Making Market maker maintains prices p t Π over n contracts Contracts are purchased in bundles r t, and q t+1 = q t + r t+1

132 Online Linear Opt. Learner maintains weights w t K over n items/experts Items have loss vector l t, cumulatively L t+1 = L t + l t+1 FTRL selects weights w t+1 = argmin w K w L t + R(w) Market Making Market maker maintains prices p t Π over n contracts Contracts are purchased in bundles r t, and q t+1 = q t + r t+1 Market maker selects prices p t +1 = argmaxp q t R(p) p Π

133 Online Linear Opt. Learner maintains weights w t K over n items/experts Items have loss vector l t, cumulatively L t+1 = L t + l t+1 FTRL selects weights w t+1 = argmin w K Learner suffers regret T t =1 w t l t min w K w L t + R(w) w L T Market Making Market maker maintains prices p t Π over n contracts Contracts are purchased in bundles r t, and q t+1 = q t + r t+1 Market maker selects prices p t +1 = argmaxp q t R(p) p Π MM has worst-case loss max p Π T t =1 ( ) p q T C(q t ) C(q t 1 )

134 Online Linear Opt. Learner maintains weights w t K over n items/experts Items have loss vector l t, cumulatively L t+1 = L t + l t+1 FTRL selects weights w t+1 = argmin w K Learner suffers regret T t =1 w t l t min w K w L t + R(w) w L T Market Making Market maker maintains prices p t Π over n contracts Contracts are purchased in bundles r t, and q t+1 = q t + r t+1 Market maker selects prices p t +1 = argmaxp q t R(p) p Π MM has worst-case loss max p Π T t =1 ( ) p q T C(q t ) C(q t 1 )

135 Let K = Π = Δ n and R(p) = An Example i p i log p i

136 Let K = Π = Δ n and An Example R(p) = i p i log p i Then there is a closed form solution for the prices/ weights: w i = exp(l i ) Σ j exp(l j ) p i = exp(q i ) Σ j exp(q j )

137 Let K = Π = Δ n and An Example R(p) = i p i log p i Then there is a closed form solution for the prices/ weights: w i = exp(l i ) Σ j exp(l j ) p i = exp(q i ) Σ j exp(q j ) randomized weighted majority / hedge

138 Let K = Π = Δ n and An Example R(p) = i p i log p i Then there is a closed form solution for the prices/ weights: w i = exp(l i ) Σ j exp(l j ) p i = exp(q i ) Σ j exp(q j ) randomized weighted majority / hedge logarithmic market scoring rule

139 More on Choosing R Interesting market properties can be described in terms of the conjugate

140 More on Choosing R Interesting market properties can be described in terms of the conjugate Worst-case market maker loss can be bounded by sup R(x) inf R(x) x Hull(ρ) x Hull(ρ)

141 More on Choosing R Interesting market properties can be described in terms of the conjugate Worst-case market maker loss can be bounded by sup R(x) inf R(x) x Hull(ρ) x Hull(ρ) Information loss (or the bid-ask spread, or the speed at which prices change) can be bounded too

142 More on Choosing R Interesting market properties can be described in terms of the conjugate Worst-case market maker loss can be bounded by sup R(x) inf R(x) x Hull(ρ) x Hull(ρ) Information loss (or the bid-ask spread, or the speed at which prices change) can be bounded too Gives us a way to optimize trade-offs in market design!

143 Example: Permutations Suppose our state space is all permutations of n items (e.g., candidates in an election, or horses in a race)

144 Example: Permutations Suppose our state space is all permutations of n items (e.g., candidates in an election, or horses in a race) Pair bets: Bets on events of the form horse i finishes ahead of horse j for any i, j Subset bets: Bets on events of the form horse i finishes in position j for any i, j

145 Example: Permutations Suppose our state space is all permutations of n items (e.g., candidates in an election, or horses in a race) Pair bets: Bets on events of the form horse i finishes ahead of horse j for any i, j Subset bets: Bets on events of the form horse i finishes in position j for any i, j Both known to be #P-hard to price using LMSR [Chen et al., 2008] The complex market framework handles both

146 Example: Permutations Subset bets ( horse i finishes in position j )

147 Example: Permutations Subset bets ( horse i finishes in position j ) Hull(ρ) can be described by a small number of constraints: price( i in slot j) =1 price( i in slot j) =1 j i

148 Example: Permutations Subset bets ( horse i finishes in position j ) Hull(ρ) can be described by a small number of constraints: price( i in slot j) =1 price( i in slot j) =1 j Easily handled i

149 Example: Permutations Pair bets ( horse i finishes ahead of horse j )

150 Example: Permutations Pair bets ( horse i finishes ahead of horse j ) Hull(ρ) is a bit uglier

151 Example: Permutations Pair bets ( horse i finishes ahead of horse j ) Hull(ρ) is a bit uglier Solution: Relax the no-arbitrage axiom

152 Example: Permutations Pair bets ( horse i finishes ahead of horse j ) Hull(ρ) is a bit uglier Solution: Relax the no-arbitrage axiom Allows us to to work with a larger, efficiently specified price space

153 Example: Permutations Pair bets ( horse i finishes ahead of horse j ) Hull(ρ) is a bit uglier Solution: Relax the no-arbitrage axiom Allows us to to work with a larger, efficiently specified price space But does it increase worst case loss?

154 Example: Permutations Pair bets ( horse i finishes ahead of horse j ) Hull(ρ) is a bit uglier Solution: Relax the no-arbitrage axiom Allows us to to work with a larger, efficiently specified price space But does it increase worst case loss? No!

155 Extensions and Additional Topics

156 Relaxing No-Arbitrage Dudík, Lahaie, and Pennock [2012] pushed on the idea of relaxing no-arbitrage to provide a general constraint generation technique for constructing efficient markets with approximately consistent prices Used this to implement the WiseQ market which allowed combinatorial bets the 2012 US presidential and senate elections [Dudík et al., 2013]

157 Continuous Outcome/Contract Spaces $0 $1 $0 0 cm 20 cm

158 Continuous Outcome/Contract Spaces $0 $1 $0 0 cm 20 cm Can discretize the outcome space ex ante, but complexity and worst-case loss grow with the number of outcomes Most early attempts to avoid ex ante discretization led to negative results [e.g., Gao and Chen, 2010] Chen, Ruberry, and Vaughan [2013] extended the duality framework to markets over continuous outcome spaces, generating markets with bounded worst case loss

159 Continuous Outcome/Contract Spaces $0 $1 $0 0 cm 20 cm Still lots of work to do here quantifying trade-offs between discretization and specially designed markets!! Worst case loss Computational complexity Granularity of predictions

160 Making a Profit / Adaptive Liquidity We have assumed that the market maker is willing to take a potential (bounded) loss in order to obtain information The ideas presented here can be modified to yield market makers guaranteed to earn a profit if the volume of trades is sufficiently high and traders disagree [e.g., Othman & Sandholm, 2011; Li & Vaughan, 2013] Yields markets with adaptive liquidity In the complete market setting, this requires that prices sum to something more than one adds some ambiguity when backing out probability estimates

161 Markets & Variational Inference The math behind these markets also parallels the math behind variational inference mean parameter prices natural parameter quantity vector sufficient statistics payoff function This connection can be used to design new scoring rules [e.g., Lahaie, working paper, 2012]

162 Price Convergence & Aggregation When do security prices converge, and do they reflect the private information or beliefs of the traders? Ostrovsky [2012] showed that prices generally converge and incorporate traders private information if traders are risk neutral and Bayesian with a common prior Price convergence also occurs for risk averse traders with heterogeneous beliefs and budgets [Sethi and Vaughan, working paper 2013] Lots to do here too! Which model(s) are reasonable? What happens in real markets?

163 See the tutorial website for additional references: